THE MAGIC SQUARE. 



59 



MAGICAL POLYGONS. 



So far we have considered only such extensions of the idea un- 

 derlying the construction of the magic square in which the figure of 

 the square was retained. We may however contrive extensions of 

 the idea in which instead of a square, a rectangle, a triangle, or a 

 pentagon, and the like, appear. Without entering into the con- 

 sideration of the methods for the construction of such figures, we 

 will give here of magical polygons simply a few examples, all sup- 

 plied by Professor Scheffler : 



i) The numbers from i to 32 admit of being written in a rect- 

 angle of 4 x 8 in such a manner that the long horizontal rows give 

 the sum of 132 and the short vertical rows the sum of 66 ; thus : 



Fig. 31- 



' 2) The numbers from i to 27 admit of being so arranged in three 

 regular triangles about a point which forms a common centre, that 

 each side of the outermost triangle will present 6 numbers of the 

 total summation 96 and each side of the middle triangle 4 numbers 

 whose sum is 61 ; as the following figure shows : 



25 



27 



2O 9 II 21 



15 l6 ' 7 8 

 ss 5 



25 

 Fig. 32. 



